Where I live, even now there are people unsure of how to pronounce some of the place names, even those less than twenty miles away. I can quite easily picture a registrar unsure of the spelling, writing it down as he thought, or one with writing that is not entirely legible - creating problems for people trying to read it. Did your mother's birth cert give any clue where the birth was registered? This could narrow down your search to a town or a county.
A list of Ulster townlands and the ones starting with T can be found at
http://www.ulsterancestry.com/ulster_tow…
Teneyueirnogue is not listed, but "ogue" is anglicization of the Irish word "óg" meaning "young" or even "little".
There is a townland in Londonderry called Termoneeny. Perhaps your townland name is a misspelling (or misreading) of Termoneeny Óg - Termoneeny Little, as opposed to a bigger place Greater Termoneeny (which I didn't find).
There is an even bigger list of Ulster townlands at
http://www.ancestryireland.com/database.…
and I went through them all, and it occurred to me that perhaps there is a passing similarity to the following townland names
http://www.ancestryireland.com/database.…
Currynierin
Tirkeeran
Tirkeeveny
Tirnageeragh
Trienaltenagh
http://www.ancestryireland.com/database.…
Tannyoky
Tirsogue
Tullynamalloge
http://www.ancestryireland.com/database.…
Finnabrogue
Tievenadarragh
http://www.ancestryireland.com/database.…
Tievaveeny
Tievenavarnog
It might be worthwhile checking the census records closest to your mother's birth year to see if the family surname is there, in any of these townlands.
Is the U in the birth cert really a V? That might indicate Tievenavarnog, in Fermanagh. That is in the parish of Drumkeeran, which appears to be a small country parish.
http://www.rootsweb.ancestry.com/~nirfer… has a link for the Tithe Applotments - if you click on the link and scroll down, there are landholders listed from 1932. Would there be a family surname showing there?
http://www.irishtimes.com/ancestor/fuses…
gives some information, including the most common names in the parish in 1862 and showing where parish records can be found (click on the "what's this?" link).
Good luck !!!